Multiplication of Fractions: Teaching for Understanding

Abstract

Multiplication of fractions is a deceptively easy skill for students to learn. In a large school district in Minnesota, fifth-grade students identified as being in the lower quartile on the district's mathematics competency test had great difficulty with all the fraction items except multiplication of fractions. Although only 18 percent of these students could find the sum of 1/2 and 1/8, 75 percent could find the product of 2/3 and 4/7. Students can successfully multiply, for example, 2/3 and 4/7, using their whole-number ideas. The answer. 8/21, can be calculated without considering the meaning of either of the fractions or of the answer.

The codevelopment of children’s fraction arithmetic skill and fraction magnitude understanding.

“…Although transfer from early understanding of fraction magnitudes (e.g., in fourth grade) to fraction arithmetic skill might happen under some circumstances, we did not find evidence that fraction arithmetic skill plays a major role in the early development of fraction magnitude understanding, or vice versa. One possible explanation for little transfer early in development and some transfer later in development is that, because a large number of U.S. children in middle childhood have little understanding of fraction magnitudes and memorize fraction arithmetic procedures (Cramer & Bezuk, 1991; Hiebert & Wearne, 1986; Siegler et al, 2011), they do not initially access either type of knowledge while learning the other type. Limited early transfer may also reflect the nature of early fraction calculation—initially, fourth graders add and subtract fractions with like denominators, which requires little understanding and a limited number of procedural rules (i.e., perform operation on numerators, but not denominators).…”

“…Most troublingly, children may not see a link between fraction arithmetic and magnitudes. Researchers have long asserted that children memorize fraction arithmetic procedures without under-standing what they mean (Cramer & Bezuk, 1991;Hiebert & Wearne, 1986). If this is true, and magnitude information is not accessed during fraction arithmetic, it is difficult to understand how transfer in either direction would occur.…”

Teacher knowledge experiment: Testing mechanisms underlying the formation of preservice elementary school teachers’ pedagogical content knowledge concerning fractions and fractional arithmetic.

“…Furthermore, common arithmetical errors and corresponding teaching approaches for addition and multiplication with fractions were shown (e.g., Mack, 1995; Siegler & Lortie-Forgues, 2015). This included appropriate representations (Cramer & Bezuk, 1991). The basic session ended with a review of the fundamental changes in conceptual understanding required to transition from the set of natural numbers to the set of rational numbers (Prediger, 2008; Siegler, Thompson, & Schneider, 2011).…”

Executable Functions of the Representations in Learning the Algebraic Concepts

“…According to Lesh (1979) and Janvier (1987b), a conceptual understanding relies on the students' experience in presenting content in each of the representational modes. Explaining Lesh's view, Cramer & Bezuk (1991) reported that mathematics understanding can be defined as the ability to represent a mathematical idea in multiple ways and to establish relationships between different modes of representation.…”